Quotients of vertex-transitive graphs

Transcription

1 Chapter 5 Quotients of vertex-transitive graphs This chapter will study some operations on vertex-transitive graphs. 5.1 Quotient graphs Let Γ = (V,E) be a digraph. For a partition B of V,thequotient Γ B is defined as the digraph with vertex set B such that, for any two vertices B,C B,B is connected to C if and only if there exist u B and v C which are adjacent in Γ. It is clear that a quotient of a connected graph is connected. Proposition 5.1 A graph is connected if and only if it is a quotient of a cycle. Let Γ = (V,E) be a vertex-transitive graph. Assume that G Aut Γ is transitive on V. If B is a G-invariant partition of V, then the quotient Γ B is called an imprimitive quotient (relative to G). In the case, Γ is called an extension of Γ B ; if in addition Γ and Γ B have the same valency then Γ is called an imprimitive cover of Γ B. Proposition 5.2 Let Γ=C n be a cycle of size n. ThenΓhas a non-trivial imprimitive quotient if and only if n is not a prime. Each of imprimitive quotients of Γ is a cycle or a single edge graph K 2. Example 5.3 Let Γ = 6K 2, 6 disjoint copies of K 2. 43

2 44 Quotients of vertex-transitive graphs There is a group X Aut Γ such that X = Z 6 Z 2 is transitive on V. There exists an X -partition B with two blocks of size 6, such that Γ B = K2. There exists a group Y Aut Γ such that Y = D 12 is transitive on V,andthere exists an X -partition B with 6 blocks of size 2 such that Γ B = C6. Let Z =A 4,actingon{1,2,3,4}. Let g = (12)(34). Then it is easily shown that Γ = Cay(Z, {g}). Thus Z acts on V regularly. Let H<Zbe isomorphic to Z 3,andlet B be the set of right cosets of H in Z.ThenΓ B = K4. Therefore, the disconnected graph Γ has connected imprimitive quotient graphs isomorphic to K 2, K 3, K 4, C 6. Exercise 5.4 Let Γ = 6K 2. Find all imprimitive quotients of Γ. 5.2 Imprimitive quotients of Cayley graphs Let Γ be Petersen graph. Then Γ can be represented as a coset graph of A 5.LetG =A 5 act on Ω = {1, 2, 3, 4, 5}. LetH= (123), (23)(45), andletg= (24)(35). Then Γ = Γ(G, H, HgH). It is known that Γ is not a Cayley graph. However, we will see that Γ is an imprimitive quotient of a Cayley graph, for example, the Cayley graph Cay(G, {g}). Theorem 5.5 Each vertex-transitive graph is an imprimitive quotient of a Cayley graph. Proof. Let Γ = (V,E) be a vertex-transitive graph, and assume that G Aut Γ is transitive on V. Then there exist a subgroup H<Gand a subset S G such that Γ=Γ(G, H, HSH). Let R be a subset of G such that S R HSH. Let Σ=Cay(G, R). Let B =[G:H]. We claim that Γ = Σ B.

3 Groups & Graphs 45 First we notice that Γ and Σ B have the same vertex set [G : H]. We need to prove that for any Hx,Hy [G : H], Hx is connected in Γ to Hy if and only if Hx is connected in Σ B to Hy. Assume that Hx is connected in Γ to Hy. By definition, yx 1 HSH, sothat yx 1 = h 1 sh 2 for some h 1,h 2 H and some s S. Thus (h 1 1 y)(h 2 x) 1 = s S R, so the vertex h 2 x is connected in Σ to the vertex h 1 1 y. Ash 2 x Hx and h 1 1 y Hy, by definition, Hx is connected in Σ B to Hy. Assume now that Hx is connected in Σ B to Hy. Then there exists some h 1 x Hx which is connected in Σ to some h 2 y Hy. By definition, (h 2 y)(h 1 x) 1 R HSH. It follows that yx 1 HSH, and hence Hx is connected in Γ to Hy. Therefore, Γ = Σ B, as desired. 2 Exercise 5.6 (1) Prove that Petersen graph is an imprimitive quotient of a connected arc-transitive Cayley graph. (2) Prove that the complete graph K 6 is an imprimitive quotient of the icosahedron.

4 46 Quotients of vertex-transitive graphs 5.3 Normal quotients Let Γ = (V,E), and assume that G Aut Γ is transitive on V. Let N be a normal subgroup of G which is intransitive on V,andletBbe the set of N -orbits in V.Then Bis a partition of V.DenotebyΓ N the quotient graph Γ B. Lemma 5.7 Using notation defined above, (1) B is a G-invariant partition; (2) for any B,C B, the vertex B is connected in Γ N to the vertex C if and only if each u B is connected in Γ to some v C. Proof. Suppose that, for some B Band some g G, B g B. Then there exist some u B and some C B such that u g C B. Hence C =(u g ) N =(u N ) g =B g, and so B is G-invariant. Assume that the vertex B B is connected in Γ N to the vertex C B.Thensome u Bis connected in Γ to v C. Since N is transitive on B, for any u B,there exists g N such that u g = u.sincegfixes C (setwise), we have v g C.Thusu is connected to v g. 2 The quotient Γ N is called a normal quotient induced by N. In particular, a normal quotient is an imprimitive quotient. The property given in Lemma 5.7 (2) is not shared by imprimitive quotient, for example, 6K 2 is A 4 -vertex-transitive, and has an imprimitive quotient isomorphic to K 4. It is known that imprimitive quotients of arc-transitive graphs are arc-transitive. For normal quotients, we have a stronger result, given in the next section.

5 Groups & Graphs Normal quotients of s-arc-transitive graphs Recall that a graph Γ = (V,E) issaidtobe(g, s)-arc-transitive if G Aut Γ is transitive on the set of all s-arcs of Γ. For example, complete graphs and hypercubes are 2-arctransitive but not 3-arc-transitive; while regular complete bipartite graphs and odd graphs are 3-arc-transitive but not 4-arc-transitive. Exercise 5.8 Let Γ be a Cayley graph of an abelian group. Prove that Γ is not 4-arctransitive unless Γ is a cycle. AgraphΓissaidtobeG-locally-primitive if G Aut Γ acts transitive on V Γand G α acts primitively on Γ(α). Lemma 5.9 Let Γ be a G-locally-primitive graph. Then for any intransitive normal subgroup N of G, eithern has exactly two orbits in V Γ and Γ N = K2,orΓis a cover of Γ N, Γ N is (G/N)-locally-primitive, and G/N Aut Γ N. Proof. Let B be the set of N -orbits in V Γ, and let K be the kernel of G acting on B. Then N K,andKæG. Thus K α æ G α,whereα VΓ. Since Γ is (G, 2)-arctransitive, G α acts 2-transitively on the neighborhood Γ(α). So K α acts on Γ(α) either trivially or transitively. Suppose that K α 1. It then follows that K α acts non-trivially on Γ(α), so that K α acts on Γ(α) transitively. Thus the valency of Γ N is equal to 1, and so Γ N = K2, which is a contradiction since N has at least 3 orbits in V. Therefore, K α =1,thatis,K acts semiregularly on V. In particular, K = N,thatis,N is the kernel of G acting on V. Hence G/N may be identified with a subgroup of Aut Γ N,and so Γ N is (G/N)-locally-primitive. 2 Perhaps the most important property of taking normal quotients of graphs is that the s-arc-transitivity of a graph is inherited by normal quotients. Theorem 5.10 (Praeger 1992) Let Γ be a connected (G, s)-arc-transitive graph for some s 2. Assume that N is a normal subgroup of G which has at least three orbits in V. Then the normal quotient Γ N is a (G/N, s)-arc transitive graph. Proof. Let B 0,B 1,...,B s and C 0,C 1,...,C s be two s-arcs of the normal quotient graph Γ N.Thenforeachiwith 0 i s 1, B i is adjacent to B i+1 and C i is adjacent

6 48 Quotients of vertex-transitive graphs to C i+1,inγ N. By Lemma 5.7, there exist u 0,u 1,...,u s and v 0,v 1,...,v s such that for each i {0,1,...,s 1}, u i B i is adjacent to u i+1 B i+1,andv i C i adjacent to v i+1 C i+1.sinceγis(g, s)-arc-transitive, there exists g G such that u g i = v i for all i {0,1,...,s}.SinceBis G-invariant, we have B g i = C i, and hence g maps the s-arc: B 0,B 1,...,B s to the s-arc: C 0,C 1,...,C s. Thus G induces a transitive action on the set of s-arcs of Γ N. By Lemma 5.9, G/N may be identified with a subgroup of Aut Γ N. 2 Exercise 5.11 Let Γ = Cay(G, S) be connected and undirected. Assume that Aut(G, S) is 2-transitive on S.Prove (i) Γ is 2-arc-transitive, (ii) all elements of S have order 2, (iii) if in addition G is abelian, then G is an elementary abelian 2-group. Theorem 5.10 suggests to study minimal s-arc-transitive graphs, that is, s-arctransitive graphs which have no non-trivial normal quotients. Let Γ be connected and undirected. Suppose that Γ is (G, s)-arc-transitive for some G AutΓ, and assume further that Γ is a minimal s-arc-transitive graph with respect to G. Then either (1) each non-trivial normal subgroup of G is transitive on V Γ, that is, G is quasiprimitive on V Γ; or (2) some non-trivial normal subgroup of G has exactly two orbits in V Γ, that is, G is bi-quasiprimitive on V Γ.

7 Groups & Graphs Quasiprimitive 2-arc-transitive graphs Let G be a quasiprimitive permutation group on Ω, and let N =soc(g), the socle of G. Let M 1,...,M k be all minimal normal subgroups of G. It is easily shown that N = M 1 M k, for some k 1. Suppose that k 2. Then since M i is transitive, we have that M i is nonabelian. Let L = M 2 M k. Then N = M 1 L. Since M 1 is transitive on Ω, for any β Ω, there exists x M 1 such that β = α x. Therefore, L β = L α x = x 1 L α x = L α, so that L α fixes all points of Ω. Hence L α =1. As Lis transitive on Ω, we obtain that L is regular. Since each M i is transitive, it follows that L = M 2 and k = 2. Similarly, M 1 is regular on Ω. Therefore, N = M 1 M 2 such that M i is nonabelian and regular on Ω; in particular, M 1 = M 2 = Ω. Since N is transitive on Ω, it follows that N α = M 1. An element of N α may be written as xx,wherex M 1 and x M 2. Suppose that for some x M 1,thereexist two different elements x,y M 2 such that xx,xy N α.thenm 2 has a non-identity element x (y ) 1 = xx (xy ) 1 N α, which is a contradiction since M 2 is regular on Ω. Further, since N α = M 1, for each element x M 1, there exists exactly one element x M 2 such that xx N α.letσbeamap from M 1 to M 2 defined: x x, where x M 1 and xx N α. Then it is easily shown that σ is an isomorphism from M 1 to M 2. In particular, M 1 = M2. Thus we have Lemma 5.12 Let G be a quasiprimitive permutation group on Ω. Then either soc(g) is a minimal normal subgroup of G, orsoc(g) =M 1 M 2 such that M 1 = M2,andthe M i are nonabelian and regular on Ω. The next lemma characterizes minimal normal subgroups of a group.

8 50 Quotients of vertex-transitive graphs Lemma 5.13 If M is a minimal normal subgroup of a group G, thenm=t 1 T k such that T 1 = =Tk is a simple group. In particular, if G is a quasiprimitive permutation group, then soc(g) =T 1 T l =T, where l 1andT 1 = = Tl is a simple group. Quasiprimitive permutation groups G are categorized into 8 types by O Nan-Scott s theorem (Praeger 1992), where N = soc(g): N is abelian (HA), N is non-abelian, N = M 1 M 2, M i is simple (HS), M i is non-simple (HC), N is a minimal normal subgroup of G, N is regular (TW), N is non-regular, N is simple (AS), N is non-simple, N α = T r (PA), N α = T r, - r =1 (SD), - r>1 (CD). A further analysis leads to the next theorem. Theorem 5.14 (Praeger 1992) Let Γ be a connected undirected graph, and assume further that G Aut Γ is quasiprimitive on VΓ and that Γ is (G, 2)-arc-transitive. Then G is of type HA, TW, AS or PA.